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Limiting cases of Nelsen's families 16 and 18

Nelsen, An Introduction to Copulas, second edition, Table 4.1 lists the limiting cases C_∞ = Π / (Σ - Π) for family 16 and C_∞ = M for family 18. Both are proved here as pointwise convergence of the CDFs on the closed unit square along any filter on which the parameter tends to +∞. The limit Π / (Σ - Π), i.e. uv / (u + v - uv), is the Clayton copula with parameter one.

For family 16 the CDF is rewritten, with q = 1/θ, as 2 / (√(T² + 4q) - T) where T = q (u + v - 1) - (1/u + 1/v - 1), which is continuous at q = 0. For family 18 one has θ / ln (e^(θ/(u-1)) + e^(θ/(v-1))) → -(1 - min(u, v)) by squeezing the logarithm between -θ/(1 - min(u, v)) and that value plus ln 2.

theorem ProbabilityTheory.Copula.tendsto_nelsen16_atTop {α : Type u_1} {l : Filter α} (θ : α → ℝ) (hθ : ∀ (a : α), 0 ≤ θ a) (hlim : Filter.Tendsto θ l Filter.atTop) (u : Fin 2 → ↑unitInterval) : Filter.Tendsto (fun (a : α) => (nelsen16 (θ a) ⋯).cdf u) l (nhds ((clayton 2 1 ⋯).cdf u))

C_∞ = Π / (Σ - Π): as θ → ∞, Nelsen's family 16 converges pointwise to the Clayton copula with parameter one, uv / (u + v - uv).

theorem ProbabilityTheory.Copula.tendsto_nelsen18_atTop {α : Type u_1} {l : Filter α} (θ : α → ℝ) (hθ : ∀ (a : α), 2 ≤ θ a) (hlim : Filter.Tendsto θ l Filter.atTop) (u : Fin 2 → ↑unitInterval) : Filter.Tendsto (fun (a : α) => (nelsen18 (θ a) ⋯).cdf u) l (nhds ((comonotonic 2).cdf u))

C_∞ = M: as θ → ∞, Nelsen's family 18 converges pointwise to the upper Fréchet bound.